On the Li-Stevi? Integral Type Operators from Weighted Bergman Spaces into β-Zygmund Spaces

For an analytic self-map ?? of the unit disk ${\mathbb{D}}$ and an analytic function g on ${\mathbb{D}}$ , we define the following integral type operators: $$T_{\varphi}^{g}f(z) := \int_{0}^{z} f(\varphi(\zeta))g(\zeta) d\zeta\quad {\rm and}\quad C_{\varphi}^{g}f(z) := \int_{0}^{z}f^{\prime}(\varphi(\zeta))g(\zeta) d\zeta$$ . We give a characterization for the boundedness and compactness of these operators from the weighted Bergman space ${L_{a}^p(dA_{\alpha})}$ into the ??-Zygmund space ${\mathcal{Z}_{\beta}}$ . We will also estimate the essential norm of these type of operators. As an application of results, we characterize the above operator-theoretic properties of Volterra type integral operators and composition operators.     

Boundary vs. interior conditions associated with weighted composition operators

Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk $\mathbb{D}$ , we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior $\mathbb{D}$ and on the boundary $\partial \mathbb{D}$ respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators.     

Reproducing Kernel Hilbert Spaces Supporting Nontrivial Hermitian Weighted Composition Operators

We characterize those generating functions ${k(z) = \sum_{j=0}^\infty z^j/\beta(j)^2}$ that produce weighted Hardy spaces H ²(β) of the unit disk ${\mathbb D}$ supporting nontrivial Hermitian weighted composition operators. Our characterization shows that the spaces associated with the “classical reproducing kernels” ${z \mapsto (1 - \bar{w}z)^{-\eta}}$ , where ${w \in \mathbb D}$ and η > 0, as well as certain natural extensions of these spaces, are precisely those that are hospitable to Hermitian weighted composition operators. It also leads to a refinement of a necessary condition for a weighted composition to be Hermitian, obtained recently by Cowen, Gunatillake, and Ko, into one that is both necessary and sufficient.     

Carleson Type Measures for Fock–Sobolev Spaces

We describe the \((p,q)\) Fock–Carleson measures for weighted Fock–Sobolev spaces in terms of the objects \((s,t)\) -Berezin transforms, averaging functions, and averaging sequences on the complex space \(\mathbb{C }^n\) . The main results show that while these objects may have growth not faster than polynomials to induce the \((p,q)\) measures for \(q\ge p\) , they should be of \(L^{p/(p-q)}\) integrable against a weight of polynomial growth for \(q<p\) . As an application, we characterize the bounded and compact weighted composition operators on the Fock–Sobolev spaces in terms of certain Berezin type integral transforms on \(\mathbb{C }^n\) . We also obtained estimation results for the norms and essential norms of the operators in terms of the integral transforms. The results obtained unify and extend a number of other results in the area.     

Weighted Composition Operators from the Minimal Möbius Invariant Space into the Bloch Space

Let ${{\varphi}}$ be an analytic self-map of the open unit disk ${{\mathbb{D}}}$ in the complex plane ${{\mathbb{C}, H(\mathbb{D})}}$ the space of complex-valued analytic functions on ${{\mathbb{D}}}$ , and let u be a fixed function in ${{H(\mathbb{D})}}$ . The weighted composition operator is defined by $$(uC_{\varphi}f)(z) = u(z)f({\varphi}(z)), \quad z \in \mathbb{D}, f \in H(\mathbb{D}).$$ In this paper, we study the boundedness and the compactness of the weighted composition operators from the minimal Möbius invariant space into the Bloch space and the little Bloch space.     

Weighted Composition Operators Between Zygmund Type Spaces and Their Essential Norms

Let ${u \in \mathcal{H}(\mathbb{D})}$ and φ be an analytic self-map of ${\mathbb{D}}$ . We estimate the essential norms of weighted composition operators uC _φ acting on Zygmund type spaces in terms of u, φ, their derivatives and the n-th power φ ⁿ of φ. Moreover, we give similar characterizations for boundedness of uC _φ between Zygmund type spaces.     

Optimal Frame Completions with Prescribed Norms for Majorization

Given a finite sequence of vectors \(\mathcal F_0\) in a \(d\) -dimensional complex Hilbert space \({{\mathcal {H}}}\) we characterize in a complete and explicit way the optimal completions of \(\mathcal F_0\) obtained by appending a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequences). Indeed, we construct (in terms of a fast algorithm) a vector—that depends on the eigenvalues of the frame operator of the initial sequence \({\mathcal {F}}_0\) and the sequence of prescribed norms—that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence \({\mathcal {F}}_0\) we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem.     

Embeddings of Müntz Spaces: Composition Operators

Given a strictly increasing sequence ${\Lambda = (\lambda_n)}$ of nonnegative real numbers, with ${\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty}$ , the Müntz spaces ${M_\Lambda^p}$ are defined as the closure in L ^p ([0, 1]) of the monomials ${x^{\lambda_n}}$ . We discuss how properties of the embedding ${M_\Lambda^2\subset L^2(\mu)}$ , where?μ is a finite positive Borel measure on the interval [0, 1], have immediate consequences for composition operators on ${M^2_\Lambda}$ . We give criteria for composition operators to be bounded, compact, or to belong to the Schatten–von Neumann ideals.     

Characterization for Fock-Type Space via Higher Order Derivatives and its Application

The space of entire functions \(\mathcal {F}_{\alpha }^{\infty }\) is mentioned in the paper of S. Janson, J. Peetre, and R. Rochberg. In this paper, we establish a characterization for the space \(\mathcal {F}_{\alpha }^{\infty }\) by \(n\) -th derivatives of entire functions. As an application of this result, we study the boundedness of Li-Stevi?’s integral operators and estimate essential norms of these operators acting on \(\mathcal {F}_{\alpha }^{\infty }\) . Furthermore we describe complete characterizations for boundedness and compactness of the Volterra-type integral operators on \(\mathcal {F}_{\alpha }^{\infty }\) .     

The Essential Norm of a Generalized Composition Operator Between Bloch-Type Spaces and Q K Type Spaces

Let ?? be an analytic self-map of the unit disk ${\rm \mathbb{D},H(\rm \mathbb{D})}$ the space of analytic functions on ${{\rm \mathbb{D}}}$ and ${g \in H(\rm \mathbb{D})}$ . We define a linear operator as follows $$C_\varphi^gf(z)=\int\limits_0^zf'(\varphi(w))g(w)\, {\rm d}w, $$ on ${ H(\rm \mathbb{D})}$ . In this paper, estimates for the essential norm of the generalized composition operator between Bloch-type spaces and Q _K type spaces are obtained.     

Volterra Type and Weighted Composition Operators on Weighted Fock Spaces

Bounded and compact product of Volterra type integral and composition operators acting between weighted Fock spaces are described. We also estimate the norms of these operators in terms of Berezin type integral transforms on the complex plan ${\mathbb{C}}$ . All our results are valid for weighted composition operators acting on the class of Fock spaces considered under appropriate interpretation of the weights.     

Parabolic Non-Automorphism Induced Toeplitz-Composition C*-Algebras with Piece-wise Quasi-Continuous symbols

In this paper we consider the C*-algebra $C^{*}(\{C_{\varphi }\}\cup \mathcal T (PQC(\mathbb T )))/K(H^{2})$ generated by Toeplitz operators with piece-wise quasi-continuous symbols and a composition operator induced by a parabolic linear fractional non-automorphism symbol modulo compact operators on the Hilbert-Hardy space $H^{2}$ . This C*-algebra is commutative. We characterize its maximal ideal space. We apply our results to the question of determining the essential spectra of linear combinations of a class of composition operators and Toeplitz operators.     

New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Let ψ and φ be analytic functions on the open unit disk $\mathbb{D}$ with φ( $\mathbb{D}$ ) ? $\mathbb{D}$ . We give new characterizations of the bounded and compact weighted composition operators W _ψ,? from the Hardy spaces H ^p , 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A _α ^p , α > ? 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal{D}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W _ψ,? f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H ¹ as well as of compactness for H ^p , 1 ≤ p < ∞, and $\mathcal{D}$ purely in terms of the symbols ψ and φ.     

Some new properties of composition operators associated with lens maps

We give examples of results on composition operators connected with lens maps. The first two concern the approximation numbers of those operators acting on the usual Hardy space H ². The last ones are connected with Hardy-Orlicz and Bergman-Orlicz spaces ${H^\psi }$ and ${B^\psi }$ , and provide a negative answer to the question of knowing if all composition operators which are weakly compact on a non-reflexive space are norm-compact.     

Compact differences of composition operators on weighted Dirichlet spaces

Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S ², the space of analytic functions whose first derivative is in H ², and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.     

Factorization Theorems for Multiplication Operators on Banach Function Spaces

Let X Y and Z be Banach function spaces over a measure space \({(\Omega, \Sigma, \mu)}\) . Consider the spaces of multiplication operators \({X^{Y'}}\) from X into the Köthe dual Y′ of Y, and the spaces X ^Z and \({Z^{Y'}}\) defined in the same way. In this paper we introduce the notion of factorization norm as a norm on the product space \({X^Z \cdot Z^{Y'} \subseteq X^{Y'}}\) that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with different factorization norms. Lozanovskii, Reisner and Maurey–Rosenthal theorems are considered in our arguments to provide examples and tools for assuring some requirements. We analyze the class \({d_{p,Z}^*}\) of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.     

Composition Operators on the Lipschitz Space of a Tree

The Lipschitz space ${\mathcal{L}}$ of an infinite tree T rooted at o is defined as the space consisting of the functions ${f : T \rightarrow \mathbb{C}}$ such that $$\beta_f = {\rm sup}\{|f(v) - f(v^-)| : v \in T\backslash\{o\}, \,v^- {\rm parent \, of \,} v\}$$ is finite. Under the norm ${\|f\|_\mathcal{L} = |f(o)|+\beta_f,\mathcal{L}}$ is a Banach space. In this article, the functions φ mapping T into itself whose induced composition operator ${C_{\varphi} : f \mapsto f \circ \varphi}$ on the Lipschitz space is bounded, compact, or an isometry, are characterized. Specifically, it is shown that the symbols of the bounded composition operators are the Lipschitz maps of T into itself viewed as a metric space under the edge-counting distance. The symbols inducing compact operators have finite range while those inducing isometries on ${\mathcal{L}}$ are precisely the onto maps fixing the root and whose images of neighboring vertices coincide or are themselves neighboring vertices. Finally, the spectrum of the operators ${C_\varphi}$ that are isometries is studied in detail.     

Schauder estimates for sub-elliptic equations

We establish Hölder estimates of second derivatives for a class of sub-elliptic partial differential operators in ${\mathbb{R}^{N}}$ of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the X _j ’s are smooth vector fields in ${\mathbb{R}^{N}}$ , and a _ij is a uniformly elliptic matrix. It is assumed that the X _j ’s satisfy homogeneity conditions with respect to a group of dilations δ _r which yield the existence of a composition law ${\circ}$ in ${\mathbb{R}^{N}}$ making the triplet ${\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}$ an homogeneous Lie group on which the X _j ’s are left translation invariant. The Hölder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on ${\mathbb{G}}$ , some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderón and Zygmund in the Euclidean setting.     

On the norms of generalized translation operators generated by Dunkl type operators

An integral representation for the generalized translation operators generated by Dunkl type operators $$ \Lambda f(x) = f'(x) + \frac{{A'(x)}}{{A(x)}}\frac{{f(x) - f\left( { - x} \right)}}{2} $$ In the spaces $ {L_p}\left( \mathbb{R} \right) $ with weight A is established, and a known estimate for their norms is improved. It is proved that under some natural assumptions on the functions A, these norms do not exceed two. Bibliography: 24 titles.